# Q查理耶多项式

q查理耶多项式是一个以基本超几何函数定义的正交多项式

${\displaystyle \displaystyle c_{n}(x;a;q)={}_{2}\phi _{1}(q^{-n},q^{-x};0;q,-q^{n+1}/a)}$

## 极限关系

${\displaystyle lim_{q\to 1}C_{n}(q^{-n};a(1-q);q)=C_{n}(x;a)}$

Q查理耶多项式之第4项(k=4):

${\displaystyle {\frac {\left(1-{q}^{-n}\right)\left(1-{q}^{-n}q\right)\left(1-{q}^{-n}{q}^{2}\right)\left(1-{q}^{-n}{q}^{3}\right)\left(1-{q}^{-x}\right)\left(1-{q}^{-x}q\right)\left(1-{q}^{-x}{q}^{2}\right)\left(1-{q}^{-x}{q}^{3}\right)\left({q}^{n}\right)^{4}{q}^{4}}{{a}^{4}\left(1-q\right)^{5}\left(1-{q}^{2}\right)\left(1-{q}^{3}\right)\left(1-{q}^{4}\right)}}}$  展开之： ${\displaystyle {\frac {1}{24}}\,{\frac {36\,nx-66\,n{x}^{2}+36\,n{x}^{3}-6\,n{x}^{4}-66\,{n}^{2}x+121\,{n}^{2}{x}^{2}-66\,{n}^{2}{x}^{3}+11\,{n}^{2}{x}^{4}+36\,{n}^{3}x-66\,{n}^{3}{x}^{2}+36\,{n}^{3}{x}^{3}-6\,{n}^{3}{x}^{4}-6\,{n}^{4}x+11\,{n}^{4}{x}^{2}-6\,{n}^{4}{x}^{3}+{n}^{4}{x}^{4}}{{a}^{4}}}}$

${\displaystyle {\frac {1}{24}}\,{\frac {{\it {pochhammer}}\left(-n,4\right){\it {pochhammer}}\left(-x,4\right)}{{a}^{4}}}}$

${\displaystyle {\frac {1}{24}}\,{\frac {nx\left(36-66\,x+36\,{x}^{2}-6\,{x}^{3}-66\,n+121\,nx-66\,n{x}^{2}+11\,n{x}^{3}+36\,{n}^{2}-66\,{n}^{2}x+36\,{n}^{2}{x}^{2}-6\,{n}^{2}{x}^{3}-6\,{n}^{3}+11\,{n}^{3}x-6\,{n}^{3}{x}^{2}+{n}^{3}{x}^{3}\right)}{{a}^{4}}}}$

## 图集

 Q-CHARLIER ABS COMPLEX 3D MAPLE PLOT Q-CHARLIER IM COMPLEX 3D MAPLE PLOT Q-CHARLIER RE COMPLEX 3D MAPLE PLOT
 Q-CHARLIER ABS DENSITY MAPLE PLOT Q-CHARLIER IM DENSITY MAPLE PLOT Q-CHARLIER RE DENSITY MAPLE PLOT